In this article, we consider complete, n-dimensional, Riemannian manifolds of finite volume. We assume that the Ricci curvature is bounded from below and normalized to have the lower bound given by Ric M ≥ −(n − 1).

For a compact Riemannian manifold, Weyl’s law describes the asymptotic behavior of the counting function of the eigenvalues of the associated Laplace operator. In this paper we discuss Weyl’s law in the context of automorphic forms. The underlying manifolds are locally symmetric spaces of finite volume. In the non-compact case Weyl’s law is closely related to the problem of existence of cusp fo...

We give bounds on finite volume expectations for a set of boundary conditions containing the support of any tempered Gibbs state and prove a theorem connecting the behavior of Gibbs states to the differentiability of the pressure for continuum statistical mechanical systems with long range superstable potentials. Convergence of grandcanonical Gibbs states is also studied. This is an extended ve...

In this paper we will show that on any complete noncompact Riemannian manifold with a finite volume there exist geodesic loops of arbitrarily small length.

Introduction We consider a Riemannian manifold M n with a positive density function Ψ (x) used to weight volume and hypersurface area. In terms of the underlying Riemannian volume dV 0 and area dA 0 , the new, weighted volume and area are given by

We show that if L is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of L are all in the interval (−δ, δ) for a fixed δ ∈ [0, 1) and no complimentary region of L is an interval bundle over a surface, then each boundary leaf of L has a nontrivial fundamental group. We also prove existence of a fixed constant δ0 > 0 such that i...

In this paper, we prove that every closed nil 3-manifold is diffeomorphic to a cusp cross-section of a finite volume complex hyperbolic 2-orbifold.

Article history: Received 25 May 2012 Received in revised form 14 September 2012 Accepted 21 November 2012 Available online 19 December 2012

Any non compact finite volume hyperbolic 3-manifold has a finite cover which admits a nondegenerate ideal triangulation. As an application, we show that the volume of those manifolds is always a critical value of a function defined from the Lobachevskii function.